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Main question: Do we need to restrict the vol-of-vol parameter in SABR further than $\text{vol-of-vol}>0$ and how do we determine the interval of vol-vol which the model is arbitragefree?

Background

Please consider a SABR model and an asset with time 0 price at $S_0=1$. Say we want the 1 year call option prices ($T= 1$) and the rate is zero $r = 0$. With the SABR parameters shown in the figure we get this:

enter image description here

The vol-vol parameter is extremely high at 7 (unlikely process, I know.) But this pricing will totally lead to Arbitrage because call with strike 1.15 is more expensive than call w. strike 110.

I have now gone through countless of papers on SABR and no-one mentions this problem. That at some point higher vol-vol might lead to arbitrage?

Info

The call option prices are computed such that

  • I have used Hagan formula to compute the implied vol
  • I have put the implied vol into the Black Scholes pricing formula as the volatility

For instance: $IV = HAGAN(k=1.15; \sigma_0,\beta,\rho,vol-vol) = 1.93$

$$BS_{call} = (\sigma = IV ; ....) = 0.6425$$

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2 Answers 2

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The SABR model itself is arbitrage-free even for high vol of vol. The question is whether the Hagan et al formula for implied volatility under the SABR model is arbitrage free - it isn't actually. For very low strikes arbitrage can occur using the Hagan et al formula for implied volatility, and perhaps also for very high vol of vol.

Question: how do you know the call with strike 115 is more expensive than the call with strike 110? The chart above only shows the IV for different strikes. Maybe you can post a table with corresponding prices.

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  • $\begingroup$ I did make an important mistake by calling the y-axis implied vol now that It actually looks like one. I have updated the graph. $\endgroup$
    – Sanjay
    Commented Jul 7, 2019 at 16:35
  • $\begingroup$ Can you verify that for this set of parameters Hagans formula is inaccurate for high strikes? $\endgroup$
    – Sanjay
    Commented Jul 7, 2019 at 16:48
  • $\begingroup$ It's safe to say something odd is going on here. It can be 1.) your implementation is incorrect or 2.) Hagan's et al formula not only breaks down, but completely disintegrates for high vol of vol. I'd check first if your implementation is indeed correct by benchmarking it against a Monte Carlo simulation for various values of vol of vol (starting at low values and progressively increasing it). But if your implementation is correct then this feature of Hagan's IV formula, which I for one wasn't aware of, is worth sharing with a wider audience. $\endgroup$
    – user34971
    Commented Jul 7, 2019 at 16:51
  • $\begingroup$ I assume that the underlying asset price is 1. So a strike of 1.07, where the inflection approximately occurs, is not that far OTM, so I find it strange that already at this strike the formula breaks. $\endgroup$
    – user34971
    Commented Jul 7, 2019 at 16:58
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I can confirm there is no error in @Sanjay graph. I obtain the same plot with Obloj correction for the SABR formula.

In fact, the popular SABR approximation formulas (Hagan or the further corrections) use as hypothesis a small vol of vol. In your case, the vol of vol $\nu$ is very large ($\nu=7$) and it is not too surprising that the approximations break down.

As mentioned by @ilovevolatility, this is not a problem of the SABR model, but of the chosen SABR approximation.

Below is an example where the arbitrage-free SABR finite difference method of Le Floc'h & Kennedy is used, and a more recent SABR approximation of Hagan (2014).

Option price with <span class=$\nu=7$ for different methods">

The Obloj formula gives essentially the same plot as yours. Here it looks flatter because of the scale. A zoom in results in the following plot enter image description here

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  • $\begingroup$ Nice, and thanks for testing for the benefit of all of us. Indeed as you say (some of) these approximations implicitly assume small vol of vol, which breaks down here. $\endgroup$
    – user34971
    Commented Aug 3, 2019 at 14:51

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